Fully-Coupled magnetoelastic model for Galfenol alloys incorporating eddy current losses and thermal relaxation

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1 Fully-Coupled magnetoelastic model for Galfenol alloys incorporating eddy current losses and thermal relaxation Phillip G. Evans, Marcelo J. Dapino The Ohio State University, 2 W 9th Ave, Columbus, OH 432 USA ABSTRACT A general framework is developed to model the nonlinear magnetization and strain response of cubic magnetostrictive materials to 3-D dynamic magnetic fields and 3-D stresses. Dynamic eddy current losses and inertial stresses are modeled by coupling Maxwell s equations to Newton s second law through a nonlinear constitutive model. The constitutive model is derived from continuum thermodynamics and incorporates rate-dependent thermal effects. The framework is implemented in -D to describe a Tonpilz transducer in both dynamic actuation and sensing modes. The model is shown to qualitatively describe the effect of increase in magnetic hysteresis with increasing frequency, the shearing of the magnetization loops with increasing stress, and the decrease in the magnetostriction with increasing load stiffness. Keywords: Galfenol, dynamic model, constitutive modeling, thermal relaxation, magnetization, magnetostriction, eddy currents. INTRODUCTION Magnetostrictive materials deform in response to magnetic fields and change magnetization in response to mechanical stress. These effects have been effective for creating sensors, actuators, and adaptive structures with high specific power and rugged operation. Galfenol is the new class of iron-gallium alloys in the Fe x Ga x (3 x 29 at%) system which exhibit moderate magnetostriction and steel-like metallurgical and mechanical properties. This unique combination of properties is expected to enable novel devices with 3-D anisotropies and responses manufactured by machining, rolling, deposition, and welding, which would also be capable of bearing compressive, tensile, bending, torsion, and shock loads. 2 Despite these advantages, dynamic devices utilizing Galfenol are expected to be heavily influenced by dynamic thermal and eddy current effects as the operating frequency is increased. 3 Fundamental models incorporating these effects are therefore needed for design, characterization, and control of future Galfenol devices. In this article we develop a fully-coupled model that characterizes the nonlinear and dynamic strain and magnetization of Galfenol in response to time-varying magnetic fields and mechanical stresses. Dynamic effects include eddy current losses, magnetic after-effects, and the mechanical dynamics of the transducer and load. The model provides a framework for characterization, design, and control of Galfenol devices subjected to combined dynamic magnetic field and stress loading. While the model is applied to Galfenol it can be directly applied to general magnetostrictive materials which exhibit cubic anisotropy. A general transducer model incorporating eddy current losses and mechanical dynamics is defined by Maxwell s equations coupled with Newton s second law. The coupling is provided by a nonlinear constitutive model relating magnetization and magnetostriction to magnetic field and stress. Other approaches to modeling dynamic effects considered in the past include coupling the classical eddy current power loss function with the Jiles-Atherton model for ferromagnetic hysteresis 4 and incorporating a special input into the Preisach model which characterizes the effect of eddy currents in terms of a step response. 5 The former is not accurate for low resistivity materials such as Galfenol because the classical power loss function assumes uniform field penetration. The latter assumes moments are either oriented up or down; this is inaccurate for Galfenol which has a significant moment population in the () plane, especially when annealed or compressed in the [] direction. This Further author information: (Send correspondence to M.J.D.) M.J.D.: dapino.@osu.edu, Telephone: P.G.E.: evans.895@osu.edu, Telephone: Behavior and Mechanics of Multifunctional and Composite Materials 28, edited by Marcelo J. Dapino, Zoubeida Ounaies, Proc. of SPIE Vol. 6929, 6929W, (28) X/8/$8 doi:.7/ Proc. of SPIE Vol W- Downloaded From: on 7/2/27 Terms of Use:

2 moment alignment results in a kinked shape of the magnetization and magnetostriction (Figure ). Dapino et al. 6 developed a dynamic actuation model for Terfenol-D incorporating mechanical actuator dynamics but not eddy current losses Magnetic Induction, T Increasing stress Increasing stress Figure. Intrinsic magnetic induction and magnetostriction measurements of Galfenol with 8.4 at.% Ga at stress levels of.38, 3.8, 4.4, 55.2, 69 MPa. 2. EDDY CURRENT LOSSES AND MECHANICAL TRANSDUCER DYNAMICS Dynamic losses due to eddy currents can be described by Maxwell s equations, E = B t, () J = E, ρ e (2) H = J. (3) A changing magnetic induction B creates an electric field E () which in turn creates a current density (2) which finally creates a magnetic field H (3) opposing the changing magnetic induction. This causes an instantaneous power loss per unit volume J E and a spatial variation in the magnetic field. Chikazumi 7 gives an approximation for this power loss for a cylindrical rod P = r2 8ρ e ( ) 2 db, dt where the magnetic induction is assumed to be uniform over the cross-section of the rod. This relation was combined with the Jiles-Atherton model of ferromagnetic hysteresis to quantify dynamic magnetic hysteresis. 4 The uniformity assumption is generally inaccurate for materials with low resistivity such as Galfenol, although it can be applied in cases when the material has sufficiently fine laminates. The spatial variation may be described by combining (), (2), and (3) into a single boundary value problem ( H) = B (H, T), (4) ρ e t H(Ω,t)=H Ω (t), (5) H(X, ) = H (X), (6) where X is the spatial location and Ω denotes the boundary. To solve the boundary value problem, a material constitutive law needs to be formulated, e.g., B(H, T) =µ [H + M(H, T)]. (7) Proc. of SPIE Vol W-2 Downloaded From: on 7/2/27 Terms of Use:

3 For magnetostrictive materials, a time-varying stress (T) results in a time-varying magnetic induction and hence eddy current losses. The presence of stress in the magnetic boundary value problem couples it to the mechanical boundary value problem defined by Newton s second law, ρ 2 u t 2 = T c u D F, t (8) u(ω,t)=u Ω (t) (9) u(x, ) = u (X) () where u is displacement, ρ is density, c D is the Kelvin-Voigt damping coefficient and F represents body forces. The strain S is the sum of the purely elastic strain obeying Hooke s law, S e = c T (where c is the compliance tensor), and the field and stress dependent magnetostriction S m (H, T). 8 For small strains S = u, the stress is T = c [ u S m (H, T)]. () The implicit relationship for stress () couples (8) to (4) and thus the coupled boundary value problem must be solved iteratively. The boundary value probelms (8) and (4) and the constitutive relationships (7) and () provide a general framework for modeling dynamic induction and strain response of magnetostrictive materials to an external field H Ω and body force F. Since the boundary value problem needs to be solved iteratively, computationally efficient constitutive relationships for M(H, T) ands m (H, T) are required. In Sections 3 and 4 a computationally efficient and 3-D constitutive model for M(H, T) ands m (H, T) incorporating rate-dependent thermal effects is developed. 3. STEADY-STATE CONSTITUTIVE MODEL Ferromagnetic materials below the Curie temperature have regions of uniform magnetization M s where the atomic magnetic moments are aligned with each other. These regions of uniform magnetization are called magnetic domains. Magnetic domains have preferred orientations which depend on the magnetic anisotropy as well magnetic field and stress. These orientations are preferred because they attain thermodynamic equilibrium. For a given magnetic field and stress, multiple preferred or equilibrium orientations may exist. In the absence of magnetic field and stress, Galfenol has six equilibrium orientations in the directions and Terfenol-D (Tb.3 Dy.7 Fe.9 ) has eight in the directions. As magnetic field and stress are applied, the equilibrium orientations rotate towards the field direction and perpendicular to the stress direction. When magnetic domains rotate, the magnetomechanical coupling energy causes a lattice strain termed magnetostriction. The macroscopic magnetization and magnetostriction are the vector sum of the domain magnetization and magnetostriction. When thermodynamic equilibrium is achieved, the steady-state macroscopic magnetization M ss and magnetostriction S m,ss of a material having r equilibrium domain orientations ˆm k is the sum of the magnetization M s ˆm k and magnetostriction Ŝ k m due to each equilibrium orientation, weighted by the volume fraction ˆξ k of domains in each orientation, M ss = M s ˆξ k ˆm k, S m,ss = ˆξ k Ŝ k m. (2) k= The equilibrium domain orientations and magnetostrictions are found from the Gibbs energy of a single domain and the equilibrium domain volume fractions are found from the Gibbs energy of a collection of domains. 3.. Equilibrium Domain Orientations The internal energy of a magnetic domain with orientation m =[m m 2 m 3 ] is due to the magnetocrystalline anisotropy energy. After considering the cubic crystal symmetry and neglecting higher order terms, the internal energy for cubic materials is 8 U (m) =K 4 (m m 2 + m 2 m 3 + m 3 m ), (3) k= where K 4 is the fourth-order, cubic anisotropy constant. The Gibbs free energy is G(H, T) =U (m) S m T µ M s m H, (4) Proc. of SPIE Vol W-3 Downloaded From: on 7/2/27 Terms of Use:

4 where T is the six-element stress vector with the first three components the longitudinal stresses and the last three the shear stresses, H is magnetic field, M s is the magnetization of a domain, and S m = S m (m) isthe magnetostriction with longitudinal components 8 and shear components S m,i = 3 2 λ m 2 i, i =, 2, 3 (5) S m,4 = 3λ m m 2, S m,5 = 3λ m 2 m 3, S m,6 = 3λ m 3 m. (6) These expressions are derived by balancing the magnetic anisotropy, elastic, and magnetoelastic coupling energies. 8 For K 4 >, the internal energy has six minima or easy axes (r =6)inthe directions and for K 4 <, the internal energy has eight minima or easy axes (r = 8)inthe directions. Applied magnetic and magnetomechanical work rotates the domains away from the easy axes towards the magnetic field direction and perpendicular to the longitudinal stress direction (see Figure 2). The equilibrium domain orientations ( ˆm k ; k =,..., r), needed for calculation of the bulk magnetization (2), are obtained through the conditions G/ m i = with the constraint m 2 +m 2 2 +m 2 3 =. The equilibrium magnetostrictions (Ŝk m; k =,..., r), needed for calculation of the bulk magnetostriction (2), are obtained by evaluating relations (5) and (6) using the equilibrium domain orientations, Ŝ k m = S m( ˆm k ) Equilibrium Domain Volume Fractions The bulk magnetization and magnetostriction due to a collection of domains is the sum of the products of the magnetization and magnetostriction equilibria with the respective volume fraction of domains ξ k in the k th equilibrium (see equation (2)). The domain volume fractions can be determined by minimization of a total Gibbs energy potential which includes the entropy of a collection of domains. The entropy of a system of N units (in this case domains) with r possible energy states has been given by terhaar 9 η = k ( ) B NV ln N!, (7) N!N 2!N 3!...!N r! where N k is the number domains in the k th energy state, N is the total number of domains, V is the control volume and k B is Boltzmann s constant. Using Stirling s approximation and ξ k = N k /N, the entropy becomes The total Gibbs energy is then given by η = k B V Ḡ(H, T) = ηθ + ξ k ln ξ k. (8) k= ξ k G k (H, T), (9) where θ is temperature and G k (H, T) =U( ˆm k ) Ŝk m T µ M s ˆm k H. The equilibrium domain configuration ˆξ k is obtained from the equilibrium conditions Ḡ/ ξ k =,constrainedto r k= ξk =, which yields k= ˆξ k = e Gk (H,T)V/k Bθ j= e Gj (H,T)V/k Bθ. (2) Proc. of SPIE Vol W-4 Downloaded From: on 7/2/27 Terms of Use:

5 y - - (c) (d) Figure 2. Gibbs energy equilibrium orientations with no applied field or stress, field and stress applied, (c) field only, and (d) stress only. Proc. of SPIE Vol W-5 Downloaded From: on 7/2/27 Terms of Use:

6 5 3 Magnetization, ka/m 5 5 Increasing stress Magnetization, ka/m Increasing bias field Stress, MPa (c) Stress, MPa (d) Figure 3. Model calculations:, inverse or actuator effect with the magnetic field applied in the [] direction at various stress bias levels; (c),(d) direct or sensor effect with the magnetic field applied in the [] direction at various magnetic field bias levels. Substitution of (2) into (2) gives the macroscopic magnetization and strain M ss (H, T) = M s ˆm k e Gk (H,T)V/k Bθ Ŝ k e Gk (H,T)V/k Bθ k= e Gk (H,T)V/k Bθ, S m,ss (H, T) = k= k= k= e Gk (H,T)V/k Bθ. (2) Model predictions for the magnetization and magnetostriction in the [] direction in response to a magnetic field at constant compressive stress are shown in Figures 3 and 3. The responses have rounded elbows and exhibit a kinked shape due to 9 degree domain orientations in agreement with the magnetization and magnetostriction measurements in Figure. Prediction of the magnetization and strain in the [] direction in response to stress at constant magnetic field is shown in Figures 3(c) and 3(d). Proc. of SPIE Vol W-6 Downloaded From: on 7/2/27 Terms of Use:

7 Magnetization, ka/m Time, ms x 6 Figure 4. Illustration of a moment jumping energy wells; step response of the macroscopic magnetization. 4. CONSTITUTIVE MODEL WITH THERMAL AFTER-EFFECTS Basso et al. have shown that rate-dependent thermal effects result in hysteresis losses in the magnetic constitutive behavior. For time-varying magnetic fields and stresses there is a delay associated with the evolution of the domain volume fractions. The volume fractions change as domain walls move; wall motion occurs as magnetic moments jump from their resident energy well to an adjacent energy well having a lower energy. A time delay occurs as moments must overcome an energy barrier separating the wells (see Figure 4). The time evolution of domain volume fractions has been expressed as a system of first-order differential equations for the uniaxial case where moments jump between the, 9, and 8 degree orientations. Extension of this approach to the general 3-D case would require differential equations describing the evolution of moments jumping between all six energy wells of the 3-D Gibbs energy (4). A more efficient form for the dynamic magnetization M and magnetostriction S m, similar to the approach of Della Torre et al., 2 is two first-order differential equations with time constant τ, driven by the steady-state constitutive models (2), τ dm dt + M = M ss, (22) M(H, T, ) = M ss (H, T), (23) τ ds m + S m = S m,ss, dt (24) S m (H, T, ) = S m,ss (H, T). (25) The step response of the magnetization is depicted in Figure 4 and Figure 5 illustrates the increase in the hysteresis of a major loop as the magnetic field frequency is increased. 5. DYNAMIC TONPILZ TRANSDUCER MODEL The general boundary value problems (4) and (8) may be simplified to provide a low-order dynamic model for a theoretical Tonpilz transducer composed of a cylindrical Galfenol rod in a closed magnetic circuit, loaded by a spring with stiffness k L, damping coefficient c L,andmassm L (see Figure 6). Since the rod is in a closed magnetic circuit with low reluctance, demagnetizing fields are negligible and the circuit can be approximated as an infinitely long rod energized by a magnetic field H ext at the surface, applied along the rod length. If the stress and strain are assumed to be uniform along the rod length, the magnetic field will also be uniform along the length and the magnetic boundary value problem reduces to a diffusion equation describing the time dependent Proc. of SPIE Vol W-7 Downloaded From: on 7/2/27 Terms of Use:

8 Figure 5. Major loop simulations with rate-dependent thermal effects. Center curve - low frequency (τf = e 6); outer curve - fast frequency (τf =.2); intermediate curve - medium frequency (τf =.). magnetic field penetration from the surface of the rod to the center, ( r H ) = B r r r ρ e t = µ ( H ρ e t + M ), t (26) H(r,t)=H ext, (27) H(r, ) = H, (28) where r is the radial location and r is the radius of the rod. If the [] crystal orientation of the Galfenol rod is oriented along the rod axis, then M/ t is given by the 3 component of the dynamic constitutive equation (22). With the uniformity assumption for stresses and strains, the mechanical boundary value problem is reduced to a second-order differential equation for the rod tip displacement from equilibrium, driven by an external force f ext at the rod tip and the average magnetostriction S m given by averaging the 3 component of S m (H(r, t),t) from (24) over the cross section of the rod, (m R + m L ) d2 u dt 2 +(c R + c L ) du dt +(k R + k L )u = EA S m (H, T ) f ext, (29) u() =, (3) where m R =(/3)ρAl is the dynamic mass of the Galfenol rod with cross-sectional area A = πr 2 and length l, c R = c D A/l is the damping coefficient of the rod, and k R = EA/l is its stiffness. The constitutive relationship for stress () reduces to T = E(u/l S m ), (3) where E is the modulus of elasticity. Figure 6. Tonpilz transducer Proc. of SPIE Vol W-8 Downloaded From: on 7/2/27 Terms of Use:

9 H ext r r ( r H ) = µ ( H r ρ e t + M t τ dm dt + M = M ss(h, T ) ) Outputs B f ext τ ds m dt T + S m = S m,ss (H, T ) Loop with convergence check S m (m R + m L ) d2 u dt 2 +(c R + c L ) du dt +(k R + k L )u = EA S m (H, T ) f ext u Figure 7. Block diagram for iterative solution of our Tonpilz transducer model. The outputs are the displacement u and the magnetic induction B averaged over the cross-section. Both are measurable quantities where u can be measured by a displacement sensor and B can be measured by a pick-up coil and integrating flux meter. The inputs are the externally applied magnetic field H ext and the externally applied force. A block diagram for the model is shown in Figure 7. The system is first discretized in space with central differences and in time with backward differences and then solved iteratively, updating the stress with each iteration. The iterations continue until a convergence criterion is satisfied. 5.. Model Simulations For all simulations the rod dimensions are /4 inches, E =6GPa,ρ =77.Kg/m 3, λ = 73e 6, λ =2e 6, µ M s =.62 T, K 4 =kj/m 3, k B θ/v = 2 kj/m 3, τ =e 9sec, and ρ e =7e 7Ωm. Fora slowly-varying external magnetic field, the magnetic field is expected to be nearly uniform over the cross-section and the average induction is expected to follow the steady-state induction B ss = µ [H ext + M ss (H ext,t)] with M ss (H, T ) given by given by (2). Furthermore, if the the spring load is zero and sufficient bias stress is applied to align domains 9 o to the rod axis, the stress should remain constant and the maximum free-strain 3/2λ should be reached. This is verified in Figure 8; Figure 8 shows that the magnetic field at all radius locations is the same; Figure 8 shows that the stress remains constant at the applied bias stress ( 5 MPa); Figure 8(c) shows that the average magnetic induction follows the steady-state induction; and Figure 8(d) shows that a rod elongation of 3/2λ is achieved. Ifaspringofstiffnessk l >> EA/l is now added, the rod displacement should be blocked and a blocking stress of 3/2λ E should be achieved in addition to the bias stress. Also, the magnetic induction should be sheared because the magnetic field has to work against the stress-induced anisotropy. This is verified in Figure 9; Figure 9 shows that the stress varies from the bias stress to the sum of the bias stress and 3/2λ E,and Figure 9(d) shows that the rod elongation is nearly zero. As the Tonpilz transducer is actuated by a dynamic external magnetic field, eddy currents give rise to magnetic fields which oppose the applied field; this results in a spatial distribution of the magnetic field. In addition, the magnetic field near the center of the rod is no longer purely sinusoidal because of the nonlinear and stress dependent permeability. Since the magnetic field near the center of the rod is delayed with respect to the external magnetic field, the average magnetic induction and the rod elongation u/l plotted against the external applied field shows significant hysteresis. These dynamic effects are shown in Figure for a Hz applied magnetic field. Proc. of SPIE Vol W-9 Downloaded From: on 7/2/27 Terms of Use:

10 Stress, MPa u/l Magnetic Induction, T B B ss 3 2 3/2λ (c) (d) Figure 8. Tonpilz model no-load simulation with a constant bias stress of 5 MPa and a mhz external field input with an amplitude of 3 ka/m. 5 Stress, MPa bias stress 3/2λ E Magnetic Induction, T B B ss u/l 3/2λ (c) (d) Figure 9. Tonpilz model blocked force simulation with k l >> EA/l, a constant bias stress of 5 MPa and a mhz external field input with an amplitude of 3 ka/m Proc. of SPIE Vol W- Downloaded From: on 7/2/27 Terms of Use:

11 .5.5 Magnetic Induction, T (c) Figure. Tonpilz model dynamic actuation simulation with k l =(EA/l)/8,abiasstressof 5 MPa and a Hz external field input where shows the spatial variation of the magnetic field with the smallest amplitude field at the center, shows the average magnetic induction vs. applied magnetic field, and (c) shows the rod elongation u/l vs. applied magnetic field Magnetic Induction, T Force, N Force, N (c) Figure. Tonpilz model dynamic sensing simulation with k l = EA/l/8, a bias stress of 2 MPa,biasfieldofkA/m and a Hz external force input where shows the eddy current induced spatial variation of the magnetic field with the largest amplitude field at the center shows the average magnetic induction vs. applied force and (c) shows the rod elongation u/l vs. applied force When the Tonpilz transducer is used as a sensor, a dynamic force input results in a dynamic change in the magnetic induction due to stress-induced domain rotation. In response to this dynamic magnetic induction, eddy currents arise which create magnetic fields in opposition to the magnetic induction change (see Figure ). This results in dynamic hysteresis loss as seen in Figure which shows the average magnetic induction vs. the force input and in Figure (c) which shows the rod elongation vs. the force input. 6. CONCLUSION A fully-coupled, dynamic model has been developed to characterize the nonlinear and dynamic strain and magnetization of Galfenol in response to time-varying magnetic fields and mechanical stresses. Dynamic effects include eddy current losses, magnetic after-effects, and mechanical dynamics of the transducer and load. The model provides a framework for characterization, design, and control of Galfenol devices subjected to combined 3-D dynamic magnetic field and 3-D stress loading. The model may be applied to magnetostrictive materials which exhibit cubic anisotropy. Dynamic hysteresis has been simulated for a -D Tonpilz transducer in both actuation and sensing modes. The Tonpilz simulation neglected demagnetization fields and flux leakage and assumes uni- Proc. of SPIE Vol W- Downloaded From: on 7/2/27 Terms of Use:

12 form stress and strain. Future work will include solving the general model for 3-D actuator geometries (with the finite element method) and incorporating demagnetizing fields and flux leakage by including Gauss s law in the magnetic boundary value problem. ACKNOWLEDGMENTS We wish to acknowledge the financial support by the Office of Naval Research, MURI grant #N4653. REFERENCES. M. J. Dapino, F. T. Calkins, and A. B. Flatau, Magnetostrictive devices, in 22nd. Encyclopedia of Electrical and Electronics Engineering, J.G.Webster,ed., 2, pp , John Wiley & Sons, A. E. Clark, J. B. Restorff, M. Wun-Fogle, T. A. Lograsso, and D. L. Schlagel, Magnetostrictive properties of body-centered cubic Fe-Ga and Fe-Ga-Al alloys, IEEE Trans. Magn. 36(5), pp , R. A. Kellogg, A. Flatau, A. E. Clark, M. Wun-Fogle, and T. Lograsso, Quasi-static transduction characterization of Galfenol, Journal of Intelligent Material Systems and Structures 6, pp , D. C. Jiles, Modeling the effects of eddy current losses on frequency dependent hysteresis loops in electrically conducting media, IEEE Transactions on Magnetics 3(6), pp , I. D. Mayergoyz, Eddy current hysteresis and the preisach model, IEEE Transactions on Magnetics 34(4), pp , M. J. Dapino, R. C. Smith, and A. B. Flatau, Structural magnetic strain model for magnetostrictive transducers, IEEE Transactions on Magnetics 36, pp , May S. Chikazumi, Physics of Magnetism, p. 32. John Wiley, New York, C. Kittel, Physical theory of ferromagnetic domains, Review of Modern Physics 2, pp , Oct D. ter Haar, Elements of Statistical Mechanics, p. 58. Butterworth-Heinemann, 3rd ed., V. Basso, C. Beatrice, M. LoBue, P. Tiberto, and G. Bertotti, Connection between hysteresis and thermal relaxation in magnetic materials, Phys. Rev. B 6, pp , Jan 2.. P. G. Evans and M. J. Dapino, State-space constitutive model for magnetization and magnetostriction of Galfenol alloys, IEEE Transactions on Magnetics, 27. In review. 2. E. D. Torre, L. H. Bennett, R. A. Fry, and O. A. Ducal, Preisach-arrhenius model for thermal aftereffect, IEEE Transactions on Magnetics 38(5), pp , 22. Proc. of SPIE Vol W-2 Downloaded From: on 7/2/27 Terms of Use: